For what I promise are un-nefarious reasons, I'd like to programmatically fetch the Ubuntu One account details for the current user of an application (eg name, email, free space). Is there way to achieve this?

Fortunately, you do not need to navigate the maze of technologies on your own (as I did), but can instead can use a handy tool Ubuntu One provides:

u1oauthrequest https://one.ubuntu.com/api/account/

If the current user doesn't have a U1 account, this will raise a CredentialsNotFound error. Otherwise it will return a server response in the form of a JSON string. Check the 'status' value to make sure the authentication went through ('200' indicates success). On success, the JSON will have the values: